Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that $x$ is a limit point of $A_1$
Let $A_m \subseteq \mathbb{R}^n$, $A_m \ne \emptyset$ and $A_{m+1} \subseteq A_m$. Suppose that $\bigcap\limits_{m=1}^{\infty} A_{m}=\emptyset$ and that $x \in \bigcap\limits_{m=1}^{\infty} \overline {A_{m}}$.
Prove that $x$ is a limit point of $A_1$.
My attempt: I'm trying ... | Hint
Deduce that $x \in A_{m+1}'$. Since $A_{m+1} \subseteq A_1$ and $x \notin A_1$, you can then deduce that $x \in A_1'$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, maximise $x+y$. The question is: if $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, what is the greatest ... | In order to get to the book's conclusion, you can test out a few numbers:
$4$ leaves remainder $4$ when divided by $6$.
$8$ leaves remainder $2$ when divided by $6$.
$12$ leaves remainder $0$ when divided by $6$.
$16$ leaves remainder $4$ when divided by $6$.
Then you can observe the possible remainders are $0, 2$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3372654",
"timestamp": "2023-03-29T00:00:00",
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Find the solution to the following differential equation: $ \frac{dy}{dx} = \frac{x - y}{xy} $ The instructor in our Differential Equations class gave us the following to solve:
$$ \frac{dy}{dx} = \frac{x - y}{xy} $$
It was an item under separable differential equations. I have gotten as far as $ \frac{dy}{dx} = \frac{... |
We write the differential equation as
\begin{align*}
xyy^\prime=x-y\tag{1}
\end{align*}
and follow the receipt I.237 in the german book Differentialgleichungen, Lösungsmethoden und Lösungen I by E. Kamke.
We consider $y=y(x)$ as the independent variable and use the substitution
\begin{align*}
v=v(y)=\frac{1}{y-x(y)}=... | {
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"timestamp": "2023-03-29T00:00:00",
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Which sets of sequence is countable and Uncountable. Consider the sequences
$$\displaystyle X=\left\{(x_n): x_n \in \left\{0,1\right\},n \in \mathbb{N} \right\}$$
$$and$$
$$\displaystyle Y=\left\{(x_n)\in X:x_n=1 \;\;\text{for at most finitely many n} \right\}$$
I have to choose which is uncountable and which is cou... | Let $A_n=\{1,\cdots,n\}$. For each $f \in \{0,1\}^{A_n}$, define $g_f:\Bbb N \to \{0,1\}$ by $$g_f(x)=\begin{cases}f(x)&\text{if}\;x \in \{1,2,..,n\}\\0&\text{otherwise} \end{cases}$$ Then each $g_f \in Y$. Then $Y$ can be written as $$Y=\cup_{n=1}^\infty Y_n$$ where $Y_n=\left\{g_f: f \in \{0,1\}^{A_n}\right\}$ . He... | {
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$(a_n)_n\subset [c,d]$ with limit point $h$ and $f: [c,d] \to \mathbb{R}$ continuous $\implies$ $f(h)$ is a limit point of $(f(a_n))_n$
$(a_n)_n\subset [c,d]$ with limit point $h$ and $f: [c,d] \to
\mathbb{R}$ continuous $\implies$ $f(h)$ is a limit point of
$(f(a_n))_n$
My attempt:
Let $f: [c,d]\to \mathbb{R}$ be... | The last sentence is not correct. Since $a_{n_k}\to h$, there is $K$ s.t. $|a_{n_k}-h|<\delta $ for all $k\geq K$. Therefore, $|f(a_{n_k})-f(h)|<\varepsilon $ when $k\geq K$, what prove the claim.
| {
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Image of a normal $*$-homomorphism Let $\mathcal M$ be a von Neumann algebra. Let $\pi:\mathcal M\to\mathcal M$ be a normal $*$-homomorphism Is $\pi(\mathcal M)$ again a von Neuman algebra? By [J. Dixmier, Les algebres d’operateurs dans l’Espace Hilbertien, 2nd ed., Gauthier-Vallars, Paris, 1969., Part I, Chapter 4.3, ... | The image $\pi(\mathcal{M})$ of a normal $*$-homomorphism $\pi\colon \mathcal{M}\to\mathcal{N}$ between von Neumann algebras $\mathcal{M}$ and $\mathcal{N}$ is indeed weakly closed in $\mathcal{N}$ (and thus a von Neumann algebra), also when $\pi$ is not injective.
In fact, one way to prove the general statement (in wh... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Number of Automorphisms of $S_6$ Trivial question about counting the number of automorphisms of $S_6$:
I know that for all $n \geq 3$, $Z(S_n)=1$, so Inn$(S_n) \cong S_n$.
I also know that $S_6$ has nontrivial outer homomorphisms, Out$S_n \cong \mathbb{Z}_2$.
Does this mean there are $S_n + \mathbb{Z}_2$ automorphisms ... | We have $\operatorname{Out}(S_6)=\operatorname{Aut}(S_6)/\operatorname{Inn}(S_6)\cong C_2$ and hence
$$
|\operatorname{Aut}(S_6)|=|S_6|\cdot |C_2|=6!\cdot 2.
$$
Here we have used that $|G/N|=\frac{|G|}{|N|}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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multiple choice question on number theory Let $a\in \mathbb{Z}$ be such that $a=b^2+c^2$ where, $b,c \in \mathbb{Z}-\{0\}$. Then $a$ cannot be written as
*
*$pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 1 \pmod4$
*$pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 3\pmod4$
*$pqd^2$ wh... | (2) Is correct since we know that every sum of two squares must be divisible by primes of the form $4n+1$ or of the form $4n+3$ if they are raised to an even exponent.
Check here for more details.
| {
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"timestamp": "2023-03-29T00:00:00",
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Proof that $3^{10^n}\equiv 1\pmod{10^n},\, n\ge 2$ This should be rather straightforward, but the goal is to prove that
$$3^{10^n}\equiv 1\pmod{10^n},\, n\ge 2.$$
A possibility is to use
$$\begin{align*}3^{10^{n+1}}-1&=\left(3^{10^n}-1\right)\left(1+\sum_{k=1}^9 3^{10^n k}\right)\\&=\left(3^{10^n}-1\right)\left(3^{9\cd... | Use induction
Basis
$$3^{100}\equiv (3^{10})^{10}$$
$$\equiv 59049^{10}$$
$$\equiv 49^{10}$$
$$\equiv 2401^5$$
$$\equiv 1\pmod {100}$$
Induction hypothesis
$$\frac{3^{10^{n}}-1}{10^n}\in\mathbb Z$$
Inductive step
$$\frac{3^{10^{n+1}}-1}{10^{n+1}}$$
$$=\frac{3^{10^{n}}-1}{10^n}\times \frac{1+\sum_{k=1}^9 3^{10^nk}}{10}$... | {
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Difference of elementary sets is elementary / Difference of intervals are intervals We call a set $E$ in $\mathbb{R}^d$ elementary iff it can be represented as a finite union of boxes. Let $E,F$ be elemnetary sets. I need to demonstrate that $E/F$ is elementary.
My question My proof (below) feels very messy and clumsy.... | Note the $C_j^c$ is a union of $2d$ unbounded boxes.
So $E_i \cap C_j^c$ is again a union of boxes since the intersection of two boxes with edges parallel to the axis is a box or the empty set.
So finally you will have two finite big unions of intersections of boxes,which is a finite union of boxes
| {
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is the quadratic equation appropriate for this? I work in a paper mill as a tech. There is a formula for percent solvents in a liquor solution. It is s=(A*P^2) + (B*P)
$S$ is the percent solvent, $A$ and $B$ are constants, $3.21953$ and $8.117$ respectively. I would like to solve for $p$, as instrumentation can tell me... | Yes, this is how it would be done.
You would then have $\displaystyle P = \frac{-B \pm \sqrt{B^2-4As}}{2A}$.
When you evaluate both the $+$ and $-$, make sure you pick the $P$ for which the percent solvent makes sense physically.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For an estimate $\hat{f}$ of a regressor $f$, showing $\mathbb{E}[(f(x)-\hat{f}(x))^2]$ is equivalent to another expression. In the context of regression for machine learning, suppose I have a function from an instance space $I$ to $\mathbb{R}$, say $f:I \rightarrow R$, and that I have an estimator $\hat{f}:I \rightarr... | Note that $\bigl(f(x)-\hat f(x)\bigr)^2=f(x)^2-2f(x)\hat f(x)+\hat f(x)^2$, which is different than what you wrote due to the sign of the last term. (EDIT: This was corrected in the question shortly after I pointed it out)
Thus,
$$
\mathbb E\bigl(f(x)-\hat f(x)\bigr)^2=\bigl(f(x)-\mathbb E\hat f(x)\bigr)^2+\mathbb E\bi... | {
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How to prove that conditional independence does not imply independence? I am trying to prove that conditional independence does not imply independence, ie that $P(A|C)P(B|C)=P(A \cap B|C) \nRightarrow P(A \cap B)=P(A)P(B)$
I guess I need a counter-example but I am struggling to find a way of homing in onto one.
So fa... | If $C = A^c,$ then as long as the $P(A) \neq 1,$ then the conditional independence equation will just be $0=0$ while we've learned nothing about whether $A$ and $B$ are independent.
That is, take any two $A, B$ that aren't independent, then $P(A) \neq 1$ (why?) and setting $C=A^c$ implies $A|C, B|C$ are independent.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\zeta(0)$ and the limit of $(1-s)\zeta(s)$ as $s\to 1$ I am hoping to compute $\zeta(0)$ where $\zeta$ is of course the Riemann zeta function. My first attempt was to use the functional equation which yields:
$$\zeta(0) = \frac{1}{\pi}\cos\left(\frac{\pi}{2}\right)\zeta(1)~.$$
Now, since $\cos(\pi/2)=0$ and $\zeta(1)\... | $\newcommand{\multichoose}[2]{{#1}^{[\!\underline{#2}\!]}}$
If you want to show
$$\lim_{s\to 1}(s-1)\zeta(s)=1\text{,}$$
here's a way that doesn't invoke other special functions, built on estimating the zeta sum by certain integrals.
Start with the rectangle rule for integration:
$$\int_{n}^{n+1}f(x)\mathrm{d}x\approx ... | {
"language": "en",
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Investigating whether a function is bounded I have the following function:
$$y=\frac{x^{2}-3}{x^{2}+7}$$
and I'm trying to determine, whether the function is bounded or not. To find the upper bound, I rewrote the function as $y=\frac{x^{2}-7+10}{x^{2}+7}$, and it's obvious that upper bound is 1.
However, how would I f... | The graph of the function is symmetric about the $y$-axis, so we only need to think about the lower bound when $x\geq 0$. And $y$ is increasing for $x\geq 0$. So the minimum occurs when $x=0$, so the greatest lower bound is $-\dfrac{3}{7}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3374245",
"timestamp": "2023-03-29T00:00:00",
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Interchange between expected value and infinite summation (Fubini theorem) Let $S_n = \sum_{i=1}^nX_i$ (where the $X_i$ are i.i.d.) and let N be a positive, integer valued r.v., independent from the sequence $X_n$.
Suppose also that $E[N]<\infty$ and $E[|X_i|]<\infty$.
What I want to prove is the following step:
$$E\bi... | I assume that all $X_i$'s are independent and have identic distribution.
In order to apply Fubini, you have to show (like you said)
$$\sum_n E(|S_n|I_{\{N=n\}}) <\infty$$
We now prove this:
$$\sum_n E(|S_n|I_{\{N=n\}}) = \sum_n E\left(\left|\sum_{k=1}^n X_k\right|I_{\{N=n\}}\right) \leq \sum_n E\left(\sum_{k=1}^n |X_k|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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In-/surjectivity of $R[X] \rightarrow \text{Map}(R,R)$ for infinite integral domain $R$. Let $R$ be a ring and consider
\begin{eqnarray*}
\phi : &R[X] &\longrightarrow \text{Map}(R,R) \\
&f &\mapsto \,\,\,(r \mapsto f(r))
\end{eqnarray*}
I have shown that $\phi$ is a ring homomorphism iff $R$ is commutative.
Now assum... | For $R$ infinite, the set of all possible functions $f:R \to R$ has cardinality $2^{\vert R \vert}$, whereas (because polynomials can be put into $1-1$ correspondence with finite sequences from $R$) $\vert R[X] \vert = \vert R \vert$. Cantor's theorem tells us that for any set, $\vert R \vert \lt 2^{\vert R \vert}$, s... | {
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"url": "https://math.stackexchange.com/questions/3374515",
"timestamp": "2023-03-29T00:00:00",
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Cities and Induction. Trying to find a dead end. There are ($n$ > 1) cities and every pair of cities is connected by exactly one road. The road can go only from A to B, only from B to A, or in both directions.
The goal is to find a dead-end city, if it exists, i.e., a city x to which there is a direct one-way road fr... | If $n=1$, ask zero questions and know that the one city $A$ is vacuously a dead end (i.e., vacuously "every other" city has a one-way road to $A$, and vacuously there is no road from $A$ to any other city).
Assume $n>1$. Pick two cities $A,B$ and ask whether there is a direct road $A\to B$.
*
*If "yes", $A$ cannot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3374598",
"timestamp": "2023-03-29T00:00:00",
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Let $\mathit f:X_1 \to X_2$ be continuous and surjective. With certain property of $d$, if $(X_1, d_1)$ is complete, then is $(X_2,d_2)$ complete? Let $\mathit f:X_1 \to X_2$ be continuous and surjective, and $d_1(p,q)\le d_2 \bigl(\mathit f(p),\mathit f(q)\bigl)$, $\forall p,q\in X_1$.
*
*If $(X_1, d_1)$ is comple... | For question 1.
Let $(y_n)\subset X_2$ be a Cauchy sequence. As $f$ is surjective, there exists $(x_n)\subset X_1$ such that $f(x_n)=y_n$ for all $n$. Now $d_1(u,v)\leq d_2(f(u),f(v))$ for all $u,v$ implies that $(x_n)$ is a Cauchy sequence. The completeness of $(X_1,d_1)$ implies the convergence of $(x_n)$ towards $x\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Handle finite integral of unbounded function I am trying to show that there exists a $\delta>0$ such that in the measure space $(X,\mathcal{A},\mu),\, u \in \mathcal{L}^1$:
$\forall E \in \mathcal{A}: \mu(E) < \delta \Rightarrow |\int_E u\,d\mu|< \frac{1}{100}$
I can show this if u is bounded. However, the problem is... | Since $u\in L^1$, $u$ is finite almost everywhere.
Therefore $|u|\land N:=min\{|u|, N\}$ monotonically increases and converges to $|u|$ a.e. as $N\rightarrow\infty$.
Choose sufficiently large $N$ so that $\int_{X}|u|d\mu-\int_{X}(|u|\land N)d\mu=\int_{X}(|u|-|u|\land N)d\mu<\epsilon$. Note that the integrand is always ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3375186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing monotone convergence of recursive relation with $ x_{n+1} = \sin(x_{n}) $ how I can show the following monotonic relation : $x_{n}-x_{n+1}>=0$ with
$x_{n+1}=sin(x_{n})$:
My Idea:
with
$x_{n}-x_{n+1}=x_{n}-sin(x_{n})=....$here I am stuck.
it were nice if someone can help me at.
greetings
| This requires an additional ssumption. For example if $x_1=-\frac {\pi} 2$ then $x_2=-1 >x_1$.
If $x_n \geq 0$ for all $x$ then this result is true and it follows from the inequality $\sin x \leq x$ for all $x \geq 0$.
Proof of $\sin x \leq x$ for $x \geq 0$: let $f(x)=x-\sin x$. Then $f(0)=0$ and $f'(x)=1-\cos x \ge... | {
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Function for finding the length of a curve connecting two points in a two-dimensional sphere I am trying to study differential geometry.
I am confused with regards to the following function for finding the length of
a curve $\gamma$ connecting two points $p, q ∈ S^2$
$$L(γ) = \int^1_0|\dot{γ}(t)| dt,γ(0) = p, γ(1) = q$... | As @math.pr said the dot shows drive function. But about formula, if you take an eleman on curve and assume to be straight, so its length can be computed by euclidean meter as below:
$$dL=\sqrt{dx^2+dy^2+dz^2} \Longrightarrow \int dL = \int \frac{\sqrt{dx^2+dy^2+dz^2}}{dt} \ dt \Longrightarrow$$
$$L = \int \sqrt{\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3375459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Prove or disprove: ${\rm Aut}(\Bbb Z_8)$ is abelian and cyclic. $\DeclareMathOperator{\Aut}{Aut}$
So for this class I was introduced to Automorphisms through the homework. We had to prove that Automorphisms under composition is a group, and the next question was asking whether $\Aut(\Bbb Z_8)$ is abelian and/or cyclic.... | Hint:
An automorphism $f$ of $\mathbf Z_8$ maps the generator $\bar 1$ onto another generator, and this image characterises $f$.
Now the generators of $\mathbf Z_8$ are $\;\{\bar 1,\bar 3,\bar 5,\bar 7\}$, hence $\operatorname{Aut}(\mathbf Z_8)$ has order $4$. Check that any automorphism $f$ satisfies $f^2=\text{id}$,... | {
"language": "en",
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In maximum likelihood estimation, why is it hard to directly optimize the likelihood function? In Boyd's Chapter 7, it writes
I am just wondering what is the reason we do not maximize the likelihood function directly and instead construts the log-likelihood function?
What is the fundamental reason that makes the produ... | In small-$n$ problems, optimizing the likelihood may be tractable, and is in practice sometimes done. However optimizing a likelihood function that involves the product of many terms (for instance $n \sim 10^8$) is computationally difficult because you must take derivatives of extremely high powers of terms and cross ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many spheres can fit inside this larger sphere? I would like to know if there is a way to do the following: calculate the maximal number of spheres of unit radius that can fit inside a sphere of radius 200 times the unit radius.
This is a generalisation of a question that was asked in a biology class. I was wonderi... | There is also a packing arrangement known as Random Close Pack. RCP depends on the object shape - for spheres it is 0.64 meaning that the packing efficiency is 64% (as you can also see in Jack D'Aurizio's link). Therefore, if the balls are randomly distributed, then you can fit approximately $0.64 \cdot \frac{\frac{4}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3375911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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how to solve $\operatorname{rem}(6^{15},17)$ without using a calculator. I am trying to solve $\operatorname{rem}(6^{15}, 17)$.
I know that we have to use congruences but don't know how to go on.
$6 ≅ 6 \mod 17$??
Can anyone please point me in the right direction?
Do I have to use CRT in here?
| A low tech solution:
$6^2 = 36 \equiv 2 \pmod{17}$ ($34$ is a multiple of $17$).
So $6^4 = (6^2)^2 \equiv 2^2 = 4 \pmod{17}$
Hence: $6^8 = (6^4)^2 \equiv 4^2 = 16 \equiv -1 \pmod{17}$
Also: $6^{15}=6^1 \cdot 6^2 \cdot 6^4 \cdot 6^8$ ($15$ is $1111$ in binary), so modulo $17$ this becomes
$6 \times 2 \times 4 \times -1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof: If $x$ is odd, then $x+2$ is odd. I'm fairly new to writing proofs so any advice can help.
I'm asked to prove the following statement: "If $x$ is odd, then $x+2$ is odd". Here is my proof:
We will prove this by contraposition: if $x+2$ is not odd, then $x$ is not odd.
Let there be an integer $k$ such that $x+... | This seems fine as long as you know that "not odd" is the same as even for integers. Also, for your opening sentence in the proof, I might say "If $x+2$ is even then we can write $x+2=2k$ for some integer $k$."
You can also just prove this directly if you know that odd integers are of the form $2k+1$. That is, if $x=2k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3376151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove or construct counterexample for statement about measure. Let (S,S,u) be a measure space and f,g $\in$ $L^0$ satisfy u(x $\in S$ : $f(x) < g(x)) > 0$. Prove or construct a counterexample for the following statement. There exists constants a, b $\in$ R s.t. u({x $\in$ S : f(x) $\leq$ a < b $\leq$ g(x)}) > 0.
My fi... | $\{x:f(x)<g(x)\}=\bigcup_{p \in Q} \bigcup_{q \in \Bbb{Q}}\{x:f(x) \leq p<q \leq g(x)\}$
There exist $p_0,q_0 \in \Bbb{Q}$ such that $u(\{x:f(x) \leq p<q \leq g(x)\})>0$
because if all these sets had measure zero then by subadditivity you would have that $u(\{x:f(x)<g(x)\})=0$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to define a group structure on a given arbitrary set? I am doing a course in Abstract Algebra, and my teacher gave me a question: "Find all possible group structures on a set X whose cardinality is ≤ 4."
I know basic group theory, but I am unable to understand what exactly does the question expect us to do (meaning... | As you are specifically wanting
*
*to understand what the question asks, and
*a hints.
I will address these and (in view of (2)) not give a worked solution.
1) As I read it, the question essentially wants you to fix a set with $4$ elements and find all group structures on this set. A different (easier) question ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $\lim_{x \to 0} \frac{(\tan(\tan x) - \sin (\sin x))}{ \tan x - \sin x}$ Find $$\lim_{x\to 0} \dfrac{\tan(\tan x) - \sin (\sin x)}{ \tan x - \sin x}$$
$$= \lim_{x \to 0} \dfrac{\frac{\tan x \tan (\tan x)}{\tan x}- \frac{\sin x \sin (\sin x)}{\sin x}}{ \tan x - \sin x} = \lim_{x \to 0} \dfrac{\tan x - \sin x}{\tan ... | @Surb identified your error with the choice$$f_1=\tan(\tan x),\,g_1=\tan x,\,f_2=\sin(\sin x),\,g_2=\sin x.$$One method that would work is to use$$\tan x=x+\frac13 x^3+o(x^3),\,\sin x=x-\frac16 x^3+o(x^3)$$together with$$x+cx^3+c(x+cx^3)^3=x+2cx^3+o(x^3),$$viz.$$\frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}=\frac{\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3376481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 3
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Confusion on integrability of stopping times I know the definition of a $\mathbb F-$stopping time $\tau$ is that for all $n \in\mathbb N $ that $\{ \tau \leq n\} \in \mathcal{F}_{n}$
How do the ideas of integrability and well-definedness of $\tau$ actually fit in to the concept of a stopping. I realize the question is ... | You should just think of $\tau$ as a random variable which takes values in $\mathbb N$ with the additional measurability property that $\{\tau \le n\} \in\mathcal F_n$ for each $n\in\mathbb N$.
Hence, $\tau$ being integrable and well-defined means the same thing as what it does for any other random variable to be integ... | {
"language": "en",
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"source": "stackexchange",
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Eigenvector of Matrix with Duplicate Columns I have this $5 \times 5$ matrix :
\begin{pmatrix}
1&1&1&1&1 \\ 2&2&2&2&2 \\ 3&3&3&3&3 \\
4&4&4&4&4 \\ 5&5&5&5&5
\end{pmatrix}
I need to find the eigenvalues and the eigenvectors. I found out that the eigenvalules are $15$ and $0$, $0$ with an algebraic multiplicity of $4$.
... | This is obviously a rank-one matrix, which you’ve verified by finding that the algebraic multiplicity of $0$ is four. Its column space (image) is spanned by $v=(1,2,3,4,5)^T$, so the only possibility for an eigenvector with a nonzero eigenvalue is a multiple of $v$.
As for the second question, remember that there’s n... | {
"language": "en",
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an ordered abelian group has no order units An element $e$ in $G^{+}$ is called an ordered unit in an ordered abelian group $(G,G^{+})$ if for any $g\in G$,there exits a positive integer such that $-ne\leq g \leq ne$.
In Rordam's book,there is an example to show that not all ordered abelian groups have order units.
He... | Suppose $f\in c_0(\mathbb N,\mathbb Z)$ is an order unit. Put $k_0=\max\{k\in\mathbb N:f(k)\neq0\}$. Define $g\in c_0(\mathbb N,\mathbb Z)$ by $g(k_0+1)=1$ and $g(k)=0$ for $k\neq k_0+1$. Then there is no $n\in\mathbb N$ such that $g\leq nf$, a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrating both sides of an equation with respect to what? Let's say we have the following DE:
$ \frac{dy}{dx} = x $
1. That's how It could be solved:
$dy = x dx$
$\int{dy}=\int{xdx}$
$y=\frac{1}{2}x^2+c$
Is it mathematically correct to separate $dy$ and $dx$ ? Or it would appear as the derivative of $y$ with respect ... | It is separable equation where $ y= y(x)$. Consider the problem
$$\frac{dy}{dx}=F(x)*Q(y)$$ where $F(x)$ depends only of $x$ and $Q(y)$ only of $y$.
If $Q(y) \neq 0$ we can rewrite this as
$$\frac{y'(x)}{Q(y(x))}= F(x)$$
$$\int_{x_0}^x{\frac{y'(t)}{Q(y(t))}dt}=\int_{x_0}^x{F(t)dt}$$
Then $y(t)=s$ We will get
$$\int_{y(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3376975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Possible for $X_{n} \to - \infty$ when $E[\xi_{i}]=0$ and $X_{n}=\sum\limits_{i=1}^{n}\xi_{i}$ I am attempting to construct an example where: $X_{n} \to - \infty-$a.s. and $E[\xi_{i}]=0$ and $X_{n}=\sum\limits_{i=1}^{n}\xi_{i}$
My idea: we would need a process that has greater weighting to the negative side, e.g.
$1-\... | Consider the random variable $\zeta_i$ with $P(\zeta_i=2^i)=2^{-i}$ and $P\left(\zeta_i=\frac{-2^i}{2^i-1}\right)=\frac{2^i-1}{2^i}$ for $i\geq 1$. Then it can be verified that $E\zeta_i=0$ and furthermore $P(\zeta_i>0\quad \text{i.o})=0$ by the Borel cantelli lemma. Hence eventually $\zeta_i<0$ with probability one so... | {
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Is there exists such figure $A$ on plane $E^2$ which isometry group is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ I suppose that there are no such $A \subset E^2$ which satisfy
$$\text{Iso}(A) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$$
But I'm stuck on showing this in formal... | One way to prove this is to prove that the isometry group of $E^2$ does not even contain a subgroup isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$. Yes, the classification of Euclidean isometries will help, but you also need to know about some special subgroups of the group of isometries. In parti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sets and subsets: What is the difference between these two statements? Would it be correct to say that $\emptyset \subseteq \emptyset$ or $\emptyset \subseteq \{\emptyset\}$? To my understanding, the null set is just an empty set, so a null set is a subset of a set that contains the null set as an element, hence $\empt... | They are both true but mean different things altogether.
$\emptyset \subset \emptyset$ is true.
It is true for any of the following reasons and maybe more.
1) Every set is a subset of itself.
2) The emptyset is a subset of any set
3) The emptyset has no elements so every element is in the emptyset is vacuuously in the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3377576",
"timestamp": "2023-03-29T00:00:00",
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Chess Board (5×5) problem
25 small squares of a 5×5 chess board are coloured with 5 different
colours available, such that each row contains all 5 available colours
and no two adjacent squares have same colour. Then the no. of
different arrangements possible are?
My attempt:
Let the colours be R,B,G,W,V
To fill... | As you noticed for the first we have $5!$ arrangement, for the second row we can use inclusion and exclusion principle as follows.
Notably, the number of cases for the second row with at least two adjacent squares with the same colour with respect to the first row, by inclusion and exclusion principle, is:
$$5\cdot 4!-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating integral of signum I am supposing to calculate the following integral: $$\int _{0}^{1}\mathrm{sgn}(x-x^{3})dx.$$ I assumed that on interval $(0,1)$ signum is positive. So:$$\int _{0}^{1}\mathrm{sgn}(x-x^{3})dx=\left [ x-x^{3}\right ]_{0}^{1}=0.$$ Is it correct?
| We have $x-x^3 >0$ for $x \in (0,1).$ Hence $\mathrm{sgn}(x-x^3)=1$ for $x \in (0,1).$
Thus $\int _{0}^{1}\mathrm{sgn}(x-x^{3})dx= \int _{0}^{1}1dx=1.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Mean value theorem for vector valued function in $\mathcal{C}([0,a]\times E\times E, E)$ Let $a$ is a real number such that $a>0$, $E$ a Banach space and $f :[0,a]\times E\times E\rightarrow E$ a continuous function.
Is the following statement correct?
For every $t\in [0,a]$, let $\overline{conv}$ the closure of the ... | In the end, you are simply integrating a continuous function. You always have
$$
\int_0^a f(s)\,ds\in a\,\overline{\operatorname{conv}}\{f(s):\ s\in [0,a]\}.
$$
This is a straightforward consequence of the definition of a Riemman integral: the Riemann sums for your integral are of the form
$$
\sum_j f(s_j)\, \Delta_j... | {
"language": "en",
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Wrong solution in commission problem. In one congress there are 15 physics and 15 math teachers. How many committees of 8 teachers can be formed with at least 4 math teachers and at least 2 physics teachers?
I know how to solve this problem. However. I can't explain why the following solution is incorrect
1)
Commissio... | You're double counting. For instance, suppose we label the math teachers $M_1$ through $M_{15}$. In the 6 mathematicians case, you're treating choosing $M_1$ as the "4" as different from choosing $M_1$ as one of the "2 remaining mathematicians from the 11 left". But there's no difference between those two cases. For th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question aboutPartial Fractions for example:
$$\frac{{{x^2} + 4}}{{x\left( {x + 2} \right)\left( {3x - 2} \right)}}\, = \frac{A}{x} + \frac{B}{{x + 2}} + \frac{C}{{3x - 2}}$$
first method is:
$${x^2} + 4 = A\left( {x + 2} \right)\left( {3x - 2} \right) + Bx\left( {3x - 2} \right) + Cx\left( {x + 2} \right)$$
but it is ... | It works from there
$$f(x)={x^2} + 4 = A\left( {x + 2} \right)\left( {3x - 2} \right) + Bx\left( {3x - 2} \right) + Cx\left( {x + 2} \right)$$
indeed
$$f(0)=4=A(2)(-2)\implies A=-1$$
and so on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3378438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Groups with the subtraction operation Why do integer mod integer sets with the operation of subtraction not form groups?
For example, integers mod 3 is {0,1,2}, which has an identity (0) and inverses (self inverses). And subtraction is an operation because any arguments into the operation outputs something still within... | Have you checked associativity?
For example, is $(2-1)-1=2-(1-1)$?
| {
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Evaluate the following infinite sum with parameter So the sum actually originates from the following integral:
$$\int_0^\infty \frac{\left\lfloor x \right\rfloor}{x^{a + 1}}dx$$
where $a$ is a real parameter. I've managed to transform the integral into the following sum (except for the case $a = 0$, which is observed s... | $HINT$
$n\frac{(n+1)^a-n^a}{n^a(n+1)^a}=\frac{n}{n^a}-\frac{n+1-1}{(n+1)^a}=\frac{1}{n^{a-1}}-\frac{1}{(n+1)^{a-1}}+\frac{1}{(n+1)^a}$
Now take cases for $a$
For $a=1$ the series diverge.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the sample quantile unbiased for the true quantile? I would like to find a way to show whether the sample quantile is an unbiased estimator of the true quantiles. Let $F$ be strictly increasing with density function $f$. I will define the $p$-th quantile for $0<p<1$ as $Q(p)=F^{-1}(p)$ and the sample quantile as $$\... | $\hat{F}_n^{-1}(p)$ is the smallest value $x$ such that at least $p$ fraction of the sample points satisfy $X_i \leq x$. In other words, at least $np$ of the sample points satisfy $X_i \leq x$, and since $np$ may not be an integer we can actually say at least $\lceil np \rceil$. Thus $\hat{F}_n(p)^{-1}=x$ if and only i... | {
"language": "en",
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The slope of the hyperbola $b^2 x^2 - a^2y^2 = a^2 b^2$ at the upper end of its right-hand latus rectum is $4/3$. What is the eccentricity? How to approach this type of problem?
The slope of the curve $b^2 x^2 - a^2y^2 = a^2 b^2$ at the upper end of its latus rectum to the right of the origin is $4/3$. What is the ecc... | As mentioned by @Blue, the latus rectum is a vertical line through the focus $(c,0) \equiv (ae,0)$.
The abcissa of the latus rectum is $ae = \sqrt{a^2+b^2}.$ So, to find the $y$ coordinates of its terminii, we have $$b^2a^2e^2 - a^2y^2 = a^2b^2$$ $$\implies y^2 = b^2(e^2-1)$$
$$\implies y' = \frac 43 = {aeb^2\over b\sq... | {
"language": "en",
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Probability that two people share the same birthday?
Suppose a room contains $n$ people. What is the probability that at least two people share the same birthday?
Let $A$ be the probability that at least two people have the same birthday. I know that the way to solve this question is actually to find the complement o... | Let $D$ denote the number of days in a year, so $D=365$ or $D=366$ (or something else) depending on how you are counting (and which planet you are living on).
The probability that no one shares the same birthday is the product of the probabilities that the second person doesn't share their birthday with the first $(D-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3379024",
"timestamp": "2023-03-29T00:00:00",
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for what a does h have an extreme point in (1,0)? I got this function:
$h(x, y) = (x − 1)^2 + 2a(x − 1)y + y^2 + y^4$
For what $a$ does h have either a maxima- or a minima-point but not a saddle point in (1,0)?
I have confirmed that the point is stationary, but it gets really tricky when trying to use the Quadratic for... | HINT
We have that
*
*$h_x=2(x-1)+2ay\implies h_x(1,0)=0$
*$h_y=2a(x-1)+2y+4y^3\implies h_y(1,0)=0$
then $(1,0)$ is a stationary point as you have noticed.
Then we need to consider
*
*$h_{xx}=2\implies h_x(1,0)=2$
*$h_{yy}=2+12y^2\implies h_y(1,0)=2$
*$h_{xy}=h_{yx}=2a\implies h_y(1,0)=2a$
finally proceed by th... | {
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"timestamp": "2023-03-29T00:00:00",
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Topology on the set $\mathbb N$, $U$ is open iff either $1\not\in U$ or else $\sum_{n\not\in U}\frac1n\lt\infty$
Define a topology on the set $\mathbb N$ of all natural numbers by calling a set $U$ open if either $1\not\in U$ or else $\sum_{n\not\in U}\frac1n\lt\infty$. Take $A=\mathbb N\setminus\{1\}$. Then, show tha... | Since $x_n$ is unbounded we can find a subsequence $x_{n_k}$ such that $x_{n_k}>k^{2}$ for all $k$. Let $U=\{1\}\cup (\{x_{n_1},x_{n_2},...\})^{c}$. Then $U$ is an open set containing $1$. Since $x_i \to 1$ we must have $x_i \in U$ for all $i$ sufficiently large but this is clearly false.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the existence of a compact subgroup Suppose $G$ is a locally Hausdorff topological group,does there must exits a non-trivial compact subgroup?
| Say that a locally compact group is topologically torsion-free if it has no non-trivial compact subgroup. Elaborating on Moishe's comment, one sees that
A locally compact group $G$ is topologically torsion-free iff $G$ is Lie, the discrete quotient $G/G^\circ$ is torsion-free, and $G^\circ$ is contractible.
[And $G^\c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving the congruence $7x + 3 = 1 \mod 31$? I am having a problem when the LHS has an addition function; if the question is just a multiple of $x$ it's fine.
But when I have questions like $3x+3$ or $4x+7$, I don't seem to get the right answer at the end.
| By Gauss's algorithm $\bmod 31\!:\,\ 7x\equiv -2\iff x\equiv \dfrac{-2}7\equiv\dfrac{-8}{28}\equiv\dfrac{-39}{-3}\equiv \,\bbox[5px,border:1px solid #c00]{13}$
Or by Inverse Reciprocity
$\bmod 31\!:\,\ \dfrac{-2}{7}\equiv \dfrac{-2-31\!\!\!\!\overbrace{\left[\dfrac{-2}{\color{}{31}}\bmod 7\right]}^{\large -2/3\,\equi... | {
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Understanding Almost Everywhere Convergence I want to better understand the following statement:
Assume $supp(f)=E=supp(f_n)$ for all $n$ with $m(E)<\infty$, $E$ measurable.
A sequence of measurable functions $\{f_n\}\rightarrow f$ almost everywhere on $E$.
*
*Does this mean:
$$\lim_{n\rightarrow\infty}f_n(x)=... | Convergence almost everywhere is not a type of convergence. We can say "$f$ converges point-wise to $g$ almost everywhere", or "$f$ converges uniformly to $g$ almost everywhere" etc. The "almost everywhere" is saying that convergence happens on the whole domain except for a set of points with measure $0$.
To give you... | {
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Semigroup isomorphism between $(\{1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$. I know that the two semigroups $(\{0,1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$ are not isomorphic because if we want to map identity elements together then it can be see that we can't have injective function between them,but wha... | Suppose there is an isomorphism $f:(\Bbb{N},+) \to ((\Bbb{N}-\{0\}, \times)$. Then since $f$ preserves idempotents, one has $f(0) = 1$. Let $a = f(1)$. Then for every $n >0$, $f(n) = a^n$. Thus $f(\Bbb{N}) = \{a^n \mid n \geqslant 0\}$ and hence $f$ is not a bijection, a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this true? If is, how can I prove this?
$ (\forall x>1, \ \exists \delta) \ \frac{e^x}{1+x^n}<\frac{e^x}{1+\delta^n}$
I use this property to prove for $a>1, \int_0^a \frac{e^x}{1+x^n}\to e-1$ but I’m not sure that why this property is true.
| Answer for the original question: $\int_0^{1} \frac {e^{x}} {1+x^{n}}dx\to \int_0^{1} e^{x} dx=e-1$ and $\int_1^{a} \frac {e^{x}} {1+x^{n}}dx\to 0$; you can apply DCT for both integrals since the integrand is dominated by $e^{x}$ which is integrable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3379930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cardinality of power set and binary sequence
Let $A$ be a set and $P(A)$ be the power set of $A$. Define $B(A)$ as
the set of all functions $F:A\rightarrow\{0,1\}$. For example,
$B(\mathbb{N})$ is the set of all binary sequences. Prove that $P(A)$
has the same cardinality as $B(A)$.
When $A$ is finite, this is ... | The bijection is given by defining the function $F_X:A\to\{0,1\}$ with $X\subseteq A$ as:
\begin{align}
F_X(a)=\begin{cases}
1&\text{if $a\in X$}\\
0&\text{if $a\notin X$}
\end{cases}
\end{align}
The things you have to show is that $G:\mathcal P(A)\to B(A)$ with $G(X)=F_X$ is a bijection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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If $\alpha,\beta\in L$ algebraic over $K$ with degrees $m,n$, then $\alpha\pm\beta$ is algebraic with degree $\leq mn$ Sorry if this is a duplicate, I couldn't find anything on here with $m,n$ not being coprime.
My attempt thus far: first observe that $[k(\alpha):k]=m$, $[k(\beta):k]=n$.
Since $(k(\alpha,\beta):k(\alp... | We are given that
$[K(\alpha):K] = m, \; [K(\beta):K] = n; \tag 1$
we observe that
$\alpha \pm \beta \in K(\alpha, \beta) = K(\alpha)(\beta); \tag 2$
using (1), by the tower law we have
$[ K(\alpha, \beta): K]$
$= [ K(\alpha, \beta):K(\alpha)][K(\alpha):K] = [ K(\alpha, \beta):K(\alpha)]m. \tag 3$
Now
$ [ K(\alpha, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Divergence of Yamabe soliton In the article
Ma, Li; Cheng, Liang, Properties of complete non-compact Yamabe solitons, Ann. Global Anal. Geom. 40, No. 3, 379-387 (2011). ZBL1225.53038. at page 382, there is a calculation. It says that if take divergence of both sides of the equation
$$\nabla^2f=Rg,$$ where $\nabla^2$ i... | First, $\nabla^k(Rg_{jk})=\nabla_jR$ because $\nabla g=0$.
Second, commuting derivatives using the definition of the Ricci tensor implies that
$$ \nabla^k(f_{jk}) = \nabla^k\nabla_j\nabla_k f= \nabla_j\Delta f + R_{jk}\nabla^k f . $$
Third, the trace of the equation gives $\nabla_j\Delta f = n\nabla_j R$.
Combining the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to Differentiate two equations to find Maximum Values I am stuck on this Differentiation problem, any help would be great!
If $A=xy$ and $x+5y=20$ find the maximum value of $A$ and the values of $x$ and $y$ for which this maximum value occurs
| Another way to handle constrained problems is the "Lagrange multiplier method. Write the function to be, in this case, maximized as $f(x,y)= xy$ and write the constraint as $g(x,y)= x+ 5y- 20$. Then $\nabla f= y\vec{i}+ x\vec{j}$ and $\nabla g= \vec{i}+ 5\vec{j}$. An extreme point, either maximum or minimum, of f wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Uniform convergence of series: $\sum_{n=1}^{\infty}{2^n\sin\left(\frac{x}{3^n}\right)}$ Task: we should find area of x>=0, that the series is Uniform convergence on this area
$$\sum_{n=1}^{\infty}{2^n\sin\left(\frac{x}{3^n}\right)}$$
There are some ways to proof "Uniform convergence of sum".
I tried to use Dirichlet a... | $$|\sin{\frac{x}{3^n}}| \leq \frac{|x|}{3^n}$$ so the series converges pointwise on the real line.
The series also converges uniformly in every bounded subset of $\Bbb{R}$ by the $M-$test of Weierstrass.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Is it true that every convex set of the Euclidean space is the sublevel set of some convex function? Let $C \subset \mathbb{R}^n$ be a convex set.
Is it true, that there exists a convex function $f$ such that
$C = \{x | f(x) \leq a\}$ for some $a \in \mathbb{R}$
| No, the claim as written is false. In dimension $n=1$, convex functions are continuous, so if $f$ is convex then $C$ would have to be closed. So as a counterexample, let $C=(-1,1)$.
(I don't know whether a slight change could fix the claim to be true.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Range of convergence of series Find all real value $a$ for which the series $$ \sum_{n=1}^{\infty}
{(\frac{1}{n} -\sin(\frac{1}{n}))^a}$$ convergent.
I tried using ratio test, logarithmic test, etc. But I could not find it.
| Since $\lim\limits_{n\rightarrow +\infty}\frac{1}{n}=0$, you have
$$ \frac{1}{n}-\sin\left(\frac{1}{n}\right)\underset{n\rightarrow +\infty}{\sim}\frac{1}{6n^3}$$
Thus
$$ \left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)^a\underset{n\rightarrow +\infty}{\sim}\frac{1}{6^an^{3a}} $$
and the series converges iff $3a>1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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What is a 3×4 coefficient matrix such that [|⃗ ] has a solution for every 3×1 vector ⃗. I know that A needs a pivot in every row and that every column vector b (with m entries) is a linear combination of the columns of A. However, I am stuck on giving an example of a matrix that would have a solution for every b vecto... | $$A=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the values of x and sum of series (as a function of x) for the geometric series which converges of $\sum_{n=0}^{\infty} (-\frac{1}{3})^n(x-7)^n$ Find the values of x.
$\sum_{n=0}^{\infty} (-\frac{1}{3})^n(x-7)^n$
I'm not sure how to combine like terms in this case but I got this:
$(-\frac{x}{3}+\frac{7}{3})^{2n}$
... | You were so close!!
Instead of saying $(\frac{-1}{3})^{n}(x-7)^n=(-\frac{x}{3}+\frac{7}{3})^{2n}$
You should have said $(\frac{-1}{3})^{n}(x-7)^n=(-\frac{x}{3}+\frac{7}{3})^{n}$
This is a property of exponents such that $a^n*b^n=(ab)^n$
The inverse of this property is useful when finding the prime factorization of pe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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RREF using mod 2 operations Can someone please help me calculate the reduced row echelon form of the following matrix: $$ \begin{bmatrix} 1&1&1&0 \\ 1&1&0&1 \\ 0&0&1&1 \end{bmatrix} \in M_{3,4}(F_2)$$
Where $F_2$ denotes the field of scalars $\{0,1\}$ with operations doen using mod $2$ arithmetic.
I am having pro... | There's no difference in the algorithm:
\begin{align}
\begin{bmatrix}
1&1&1&0 \\
1&1&0&1 \\
0&0&1&1
\end{bmatrix}
&\to
\begin{bmatrix}
1&1&1&0 \\
0&0&1&1 \\
0&0&1&1
\end{bmatrix} && R_2\gets R_2+R_1
\\[2ex]&\to
\begin{bmatrix}
1&1&1&0 \\
0&0&1&1 \\
0&0&0&0
\end{bmatrix} && R_3\gets R_3+R_2
\\[2ex]&\to
\begin{bmatrix}
1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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HHT and HTH in tossing a coin A coin is flipped infinitely until you or I win. If at any point, the last three tosses in the sequence are $HHT$, I win. If at any point, the last three tosses in the sequence are $HTH$, you win. Which sequence is more likely?
Unfortunately, this configuration does not seem like the ones ... | Hm, I think I actually have an answer. Assuming there is no TT, consider the first occurence of HTH and suppose it is winning. Then just before it we can't have HH (else HHHTH has HHT in the beginning), we can't have TT by above, we can't have HT (else we get TT) and if we have TH, then in THHTH we have HHT in the midd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Solve $x-1 \ge \sin(x)$ I got as far as $-1 = \sin(x)-x$.
I don't know what to do next. Pretty sure I am forgetting some simplification rule.
| Others have suggested there is no analytical solutions, however;
Let $x=\frac{5\pi }{2}$, then $\frac{5\pi }{2}-1\geq \sin(\frac{5\pi }{2})=1$. Just an example, you can go from here.
It seems there are infinite solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Do any two affine rotations with no common fixed point generate an infinite group? Assume we have two affine rotations of the plane around two different fixed points.
Do they generate an infinite group?
| If you have a group $G$ of affine self-transformations of a vector space $V$: then if $G$ is finite then $G$ fixes $\frac1{|G|}\sum_{g\in G}g(0)$.
By contraposition, if $G$ is generated by a subset $S$ and there is no common fixed point for elements of $S$, then $G$ is infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What's the difference between deciding if a mathematical statement is true VS proving it? Isn't a statement only true if you can prove it?
Edit: To elaborate, when reading about foundations of math, there seems to be concepts of completeness and decidability that seems to suggest they are proving and deciding if true a... | My opinion is that there is sometimes theorems that work everytime but we still don't have a specific reasonable proof for (and they are quite rare). Also there is something called axioms that are agreed on to be true with no proof, just by convention (I know this is a little it bit different than "deciding" if it is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can a "proper" function have a vertical slope? Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us:
Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails the vertical line test... |
My question is: how can we have a function that also has a vertical
tangent? To get a vertical tangent we need 2 vertical points...
As others have pointed out, this is the crux of the misunderstanding. That said, I'd like to try and succinctly highlight the core issue: and that is that derivatives are not defined by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3381871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
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"answer_id": 9
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What is signified by the use of a "big" integral sign? [photo example] I encountered what I'll call, for lack of a better term, a "big" integral sign in a generic form of Integral Product Rule. I call it "big" relative to those preceding, and especially to the one immediately following it, in this example. This notatio... | It is the integration-by-parts formula, although demonstrated in a quite confusing way. Especially, it is unclear which function $\int$ is applied to. Using parentheses to emphasize the scope of $\int$ for each instance, we may instead write
$$\int(fg) = \left(\int f\right) g - \int \left(\left( \int f \right) g'\right... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about how to define open sets for continuous rational functions I am having difficulty doing the following question using the language of open sets.
Let $(X,\mathcal{T})$ be a topological space, and let $f,g:X\rightarrow \mathbb{R}$ be continuous functions.
Let $A=\{x\in X:g(x)=0\}.$ Prove that the function... | As restrictions of continuous functions are continuous,
f and g over R - A are continuous.
As g is never 0 over R - A, 1/g is defined and continuous over R - A.
Since the product of two continuous functions is continuous,
f/g is continuous over R - A.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the volume of the function $y=\frac{6}{x}$ Consider the function $y=\frac{6}{x}$ bounded by $y=0$, $x=1$,and $x=3$ and is rotated around the $x$-axis.
Using the disk method I setup the integral as
$$\pi\int_1^33^2-\left(\frac{6}{x} \right)^2dx $$
solving gave me $42\pi$ but the answer should be $24\pi$, my questi... | The integrand is incorrect. It should be just $(\frac{6}{x})^2$. There's no reason to subtract this from $3^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\sqcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N} $ My Quesion: Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\bigcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N}$, where $[k]$ means the integral pa... | "You already provided a complete solution to your problem" Yes, I just proved it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Find the probability mass function of X An airline operates a small 10-seat aircraft. It has just made 12 reservations for the next flight: the first 7 bookings will be confirmed at takeoff. Of the 5 other bookings, each of the bookings will be confirmed with probability = 1/2 and independence.
*
*What is the probab... | The number of people that arrive is given by $7+W$ where $W\sim\mathrm{Bin}(5,1/2)$. We want the probability $\mathbb P(7+W>10)=\mathbb P(W>3)$. We compute this by
\begin{align}
\mathbb P(W=4)+\mathbb P(W=5) &= \binom 54(1/2)^5 +\binom 55(1/2)^5 \\
&= 6(1/2)^5\\
&= 3/16.
\end{align}
$X$ is simply $(W-2)^+:= \max\{W-2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$Y = |X|$, where $X \sim \text{N}(\mu, \sigma^2)$: How does the PDF and CDF change? I just encountered the random variable $Y = |X|$, where $X \sim \text{N}(\mu, \sigma^2)$. Now, based on what we know about the absolute value function, this random variable is still continuous; however, the absolute value function means... | $$P(\lvert X\rvert\le\alpha)=\begin{cases}P(-\alpha\le X\le\alpha)&\text{if }\alpha>0\\ 0&\text{if }\alpha<0\end{cases}$$
Therefore the cdf is $$F_{\lvert X\rvert}(\alpha)=\begin{cases}0&\text{if }\alpha<0\\ F_X(\alpha)-\sup_{\beta<-\alpha} F_X(\beta)&\text{if }\alpha\ge 0\end{cases}$$
Since $F_X$ is continuous, $F_{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove convergence of two series: I would like to prove if the next two series are convergent.
First:
$$
\sum_{n=1}^{\infty}\log\left(\frac{n+1}{n}\right)\arcsin \left(\frac{1}{\sqrt{n}}\right)
$$
I think that this series is convergent, so $$\arcsin\left(\frac{1}{\sqrt{n}}\right)$$ is similar to $$\frac{1}{\sqrt {n}}$$... | Answer for the second series: this series converges absolutely if $\sum \frac {1-\cos(\frac 1 n)} {|\cos(\frac 1 n)|}$ converges. Since the denominator tends to $1$ it is enough to prove convergence of $\sum {(1-\cos(\frac 1 n))}$. This series converges because $1-\cos \theta \leq \frac {\theta^{2}} {2}$ and $\sum \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Division by $dx$ in multi-variable calculus .... I am stuck on this doubt :
Suppose $f=f(x,y,z).$ Hence, $ df= \frac {\partial f}{\partial x}dx + \frac { \partial f}{\partial y}dy + \frac {\partial f}{\partial z}dz.$
Then, is the following equation correct :
$$\frac {df}{dx}=\frac {\partial f}{\partial x}+\frac {\part... | If $$f=f(x,y,z)$$ where $x,y,z$ are independent variables, then you may divide $df$ by $dx$.
For example $$f(x,y,z)= xyz+x^2+y^2+z^2$$
$$df = (yz+2x)dx + (xz+2y)dy + (xy+2z)dz$$
$$\frac {df}{dx} = yz+2x + (xz+2y)\frac {dy}{dx} + (xy+2z)\frac {dz}{dx} =yz+2x $$
Which is the same thing as $\frac {\partial{f}}{\partial ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Green's function for Laplace Equation and the unit ball In order to find a solution to
$\Delta u(x) =0$ in $B(0,1)$ and $u(x)=g(x)$ on $\partial B(0,1)$
the book I am reading ((Graduate Studies in Mathematics) Lawrence C. Evans - Partial Differential Equations_ Second Edition -AMS (2010) Page 39) uses the Greens func... | Although the inversion is not possible, $G$ can be extended to $x = 0$, provided $y \neq 0$. This follows by observing that $|x|\tilde{x}$ has norm one for every $x \neq 0$ and upon recalling that the function $\Phi$ is radial. To be precise, for every sequence $x_n \to 0$, $x_n \neq 0$, we have that $$\lim_n \left[\Ph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Quadrilateral ABCD is inscribed in circle, $AB=4, BC=5, CD=6, DA=7$, how long is $AC$? Quadrilateral ABCD is inscribed in circle, $AB=4, BC=5, CD=6, DA=7$, how long is $AC$?
I think I'm probably supposed to use Ptolemy's to solve this, but I don't know if it's possible. Is there a way to do this problem using Ptolemy's... | By the theorem of cosines we get
$$AC^2=4^2+5^2-2\times 4\times 5\cos(\beta)$$
$$AC^2=7^2+6^2-2\times7\times6\cos(180^{\circ}-\beta)$$ and $$\cos(\pi-x)=-\cos(x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the idea behind expm1 to avoid cancellation error? It is well known that when x is close to 0, computing exp(x) - 1 introduces significant cancelation errors. As such, we have expm1 implemented in c99 and python.
My question is how expm1 avoids cancellation error? Can anyone give me a general idea without too ... | Around $x=0$ the exponential is computed as some version of $1+xp(x)$ or $\frac{1+xg(x^2)}{1-xg(x^2)}$ or similar with some polynomials that approximate the exact term behind them in an uniform fashion on some interval.
For other values the logarithm laws $e^x=(e^{x/2^m})^{2^m}$ or $e^x=2^k3^me^{x-k\ln2-m\ln3}$ are us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine the intersection of equivalence relations Let $X$ be a set. We consider the relations on $X$ as subsets of $X\times X$. Let $U\subseteq X\times X$ be a subset, and let $S_U$ be the set of all equivalence relations on $X$ that contain $U$ as subset.
Let $$R:=\bigcap_{S\in S_U}S$$ which is an equivalence relati... | Well, $R$ is always going to be an equivalence relation and we can use this to help us.
Note that $(x,|x|+k)\in U$ for any $x\in\mathbb{Z}$ and any $k\geq 100$. Hence, if we let $x,y\in \mathbb{Z},$ then $(x,|x|+|y|+100)\in U$ by the previous and, likewise, $(y,|x|+|y|+100)\in U.$ Accordingly, these elements are also i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Modelling a horizontal mass damper using differential equations I am trying to solve this second-order differential equation:
$y'' + y'+ y + (y')^2 =0$
I was able to solve the equation $y'' + y'+ y $, by substituting $y$ as $Ae^{kt}$.
But now I have this new term $(y')^2$.
Note: This equation represents the simplifi... | Let $u=\dfrac{dy}{dt}$ ,
Then $\dfrac{d^2y}{dt^2}=\dfrac{du}{dt}=\dfrac{du}{dy}\dfrac{dy}{dt}=u\dfrac{du}{dy}$
$\therefore u\dfrac{du}{dy}+u^2+u+y=0$
$(y+u^2+u)\dfrac{dy}{du}=-u$
This belongs to an Abel equation of the second kind.
Let $v=y+u^2+u$ ,
Then $y=v-u^2-u$
$\dfrac{dy}{du}=\dfrac{dv}{du}-2u-1$
$\therefore v\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove there are distinct $x_1,\,x_2,\cdots,\,x_n$ such that $ \sum_{i=1}^n\frac{p_i}{f'(x_i)}=\sum_{i=1}^n p_i. $
Suppose $f(x)$ is differentiable on $[0,\,1]$, $f(0)=0$, $f(1)=1$ and $p_1,\,p_2,\cdots,\,p_n$ are $n$ positive real numbers. Prove there are distinct $x_1,\,x_2,\cdots,\,x_n$ such that
$$
\sum_{i=1}^n\f... | Proof. $\blacktriangleleft$ Assume $\sum p_j = 1$, otherwise replace $p_j$ by $p_j/p$ for each $j$. By continuity and the Intermediate Value Theorem, there is some $y_1 \in (0,1)$ that $f(y_1) = p_1$, then there is some $y_2 \in (y_1, 1)$ that $f(y_2) = p_1 + p_2$. Do this $n-1$ times, we obtain that
$$
0 = y_0 < y_1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\left ( 1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}<2$ Prove that $$\left ( 1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}<2 \tag{1} $$ $\forall$ $n \gt 1$
I tried using Induction:
For th... | Note that by Bernoulli inequality in the form
$$(1+x)^a<1+ax, \quad 0<a<1$$
which can be easily proved by induction, we obtain
$$\left ( 1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}<1+\frac{n^{\frac{1}{n}}}{n^2}+1-\frac{n^{\frac{1}{n}}}{n^2} =2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3383865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Boundary of a compact connected set in $\Bbb R^2$ If I have a compact connected set in $\Bbb R^2$, and I'm examining the boundary points of this set. Is it true that around every cusp/corner on the boundary, there's an open interval where the boundary is smooth? This seems intuitive to me, but I don't know if it's true... | Take a function $f : [0,1]\to \mathbb R$ which is continuous but nowhere differentiable with $f\geq 0, f(0)=f(1)=0$, and take $\{(x,y) \mid x\in [0,1] \land -f(x)\leq y \leq f(x)\}$.
This is clearly connected and compact. But the boundary is not smooth at any point, as the boundary is precisely the union of the graphs ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Will the graphs of equivalent indefinite integrals always look identical? I came across the following indefinite integral
$$\int {\sqrt{1-\sin{\left(2x\right)}}}\space \mathrm{d}x \space (0 \leq x \leq \pi )$$
I attempted to solve it as follows:
$$ u = 1 - \sin\left(2x\right) \implies \sin\left(2x\right) = 1- u$$
$$\... |
Note that the integration is over a periodic function with periodic non-differentiable points as shown in the plot. Follow the steps below to perform such indefinite integration.
$$I=\int {\sqrt{1-\sin{\left(2x\right)}}}\space \mathrm{d}x
= \int {\sqrt{(\sin{x} - \cos{x})^2 }}\space \mathrm{d}x $$
$$= \int |\sin{x} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Volume integration Let
$$D = \left\{(x,y,z)\in\mathbb{R}^{3}\mid x\ge0,0\le y\le x, x^2+y^2\le {16}, 0\le z\le {5}\right\}.$$
I want to integrate
$$\displaystyle\iiint\limits_{D}\left({-4\,z+y^2+x^2}\right)\,\mathrm{d}V $$
We can see that $x^2+y^2=r^2$ so $r^2=16$. $r\to[0,16]$ and $\theta\to[0,2\pi]$ and $z \to[0,5]$
... | There are two errors.
*
*$r^2=16$ means $r=4$ not $16$
*$x\ge 0$ and $0\le y\le x$ means that $\theta$ varies between $0$ and $\pi/4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Decomposition of Bilinear Form Let $E=\text{span}\{x_{1},...,x_{n}\}$ and $F=\text{span}\{y_{1},...,y_{n}\}$. Assume that $B$ is a bilinear form on $E\times F$, the author claims that one can write
\begin{align*}
B(x,y)=\sum_{j=1}^{m}\theta_{j}(x)\omega_{j}(y),
\end{align*}
where $\theta_{j}\in E^{\#}$, $\omega_{j}\in... | Consider the following : for each $(i,j)$, put $\theta_{(i,j)} = B(e_i, f_j) e_i^*$ (where $e_i^*(\sum_k \lambda_k e_k ) = \lambda_i$) and $\omega_{(i,j)} = f_j^*$.
Then compute $\sum_{(i,j)} \theta_{(i,j)}\omega_{(i,j)} (x,y) = \sum_i \sum_j e_i^*(x)f_j^*(y)B(e_i,f_j)$, so with your notations, $\sum_{(i,j)} \theta_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding $\sin 0.01$ to a first-order approximation (in the sense of a Taylor series expansion around $0$) I am trying to understand what I need to calculate here exactly:
To a first-order approximation (in the sense of a Taylor series expansion around 0), what is $\sin 0.01$?
If I understood it correctly, I have to c... | You're exactly right (in answer)! You should be expecting this because of the so-called Small Angle Approximation that $\sin x \approx x$ when $x \approx 0$. Then as $0.01 \approx 0$ we have $\sin(0.01) \approx 0.01$, whatever that all means.
Note however that the first order approximation is
$$
T_1(x)= f(a) + f'(a)(x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
To use Mean Value Theorem to prove $f(x)=\tan(x)$ increases over $(-\pi/2,\pi/2)$, don't we need $f(\pm\pi/2)$? Yet these values are undefined.
Prove with the Mean Value Theorem that the function $\tan(x)$ increases in the interval $(\frac{-\pi}{2}, \frac{\pi}{2})$.
My problem is that to use the Mean Value Theorem yo... | We have $\tan(x) = \frac{\sin(x)}{\cos(x)}$, hence $\tan'(x) = \frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)} = \frac 1{\cos^2(x)}$. Let $x,y\in (-\pi/2,\pi/2)$, $x<y$. Then there is some $\xi\in (x,y)$ such that
$$
\tan(y)-\tan(x) = \tan'(\xi)(y-x) = \frac{y-x}{\cos^2(\xi)}> 0.
$$
Hence, $\tan(x)<\tan(y)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Showing $x^4+x^3+2x+15$ is irreducible in $\mathbb{Q}[x]$ Specifically, I'm trying to solve this problem:
Prove that $p(x)=x^4+x^3+2x+15$ is an irreducible polynomial in $\mathbb{Q}[x]$ by considering $p(x)$ mod $3$ and showing that $p(x)$ has no rational roots.
I'm able to show this is irreducible by applying the ra... | If I recall my algebra correctly, there's a theorem that says that if $p(x) \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}[x],$ then it's irreducible over $\mathbb{Q}[x].$ Therefore, if $p(x)$ has no rational roots but is reducible over $\mathbb{Z}[x],$ then it'll be the product of two quadratics with integer coef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Units in $R[x,x^{-1}]$ Let $R$ be an integral domain. I am looking for a general way to describe the units in
$$ R[x,x^{-1}] := R[x,y]/(xy-1).$$
Clearly
$$\{rx^n \mid n \in \mathbb{Z}, r \in R^\times \} \subseteq R[x,x^{-1}]^\times$$
is a subgroup, but how do I know whether it's all? I was trying to argue with degrees... | When $R$ is not integral it's not necessarily true that this is all there is : look at $R= \mathbb Z/4$ and $(2x+1)^2 = 4x^2+4x+1 = 1$ : even in $R[x]$ there can be other units.
If $R$ is an integral domain, those are indeed the only ones : take $fg = 1$, let $k$ (resp. $j$) be the highest index for which $f_k$ (resp.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Expected Number of rolls of 1 before first 6 is rolled Say we have a fair six-sided die, and we want to find the expected number of 1s rolled before the first six is rolled.
Apparently this can be solved by conditioning/recursion. I know that by that same method there are 5 expected rolls before rolling a 6 (with roll... | If you know the number of expected rolls before first rolling a $6$, then this is the expected number of $1$s before the first $6$ plus the expected number of $2$s before the first $6$ plus ... plus the expected number of $5$s before the first $6$. But each of these is equal, by symmetry, so you just need to divide by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Determine $x+y$ where $x$ and $y$ are real numbers such that $(2x+1)^2+y^2+(y-2x)^2=\frac{1}{3}$ Determine $x+y$ where $x$ and $y$ are real numbers such that $(2x+1)^2+y^2+(y-2x)^2=\frac{1}{3}$
I used the quadratic equation to get $$x=\frac{y-1\pm\sqrt{-2y-3y^2-\frac{5}{3}}}{4}$$
But I don’t see how that helps, hints a... | Hint: Show that $\frac{1}{3}$ is the unique minimum of the function $f(x,y)=(2x+1)^2+y^2+(y-2x)^2$. Then find the $(x,y)$ where this minimum occurs.
Your expression under the radical is also incorrect. It should be $-3y^2-2y-\frac{1}{3}$, or $-\frac{1}{3}(9y^2+6y+1) = -\frac{1}{3}(3y+1)^2$. Thus there is exactly one va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Approximation near singularity of $1/\sin$ I'm looking for an approximation $f(x)$ of $\frac{1}{\sin(x)}$ near the singularity at $x=0$.
Can you propose a function or literature or a key word, which leads me to $f(x)$? $f(x)$ must not have a singularity at $x=0$ and needs to be continous.
| Multiply by anything that is close to $1$ far from $x=0$ and has a minimum at $(0,0)$.
Like
$$\frac{x^2}{(x^2+\epsilon)\sin(x)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
In how many ways can 10 blankets be given to 3 beggars such that each recieves at least one blanket? The question was to find the number of ways in which 10 identical blankets can be given to 3 beggars such that each receives at least 1 blanket. So I thought about trying the multinomial theorem...this is the first time... | Try stars and bars. You have $10$ stars for the $10$ blankets:
$**********$
Now you can use $2$ bars to split this into $3$ sections. For example
$**|*******|*$
would mean beggar $1$ gets $2$ blankets, beggar $2$ gets $7$ blankets, and beggar $3$ gets $1$ blanket
Since each beggar should get at least $1$ blanket, we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Geometry question involving the length of a chord Two chords $AB$ and $AC$ are drawn inside a circle with diameter $AD$. The angle $BAC = 60$, $AB = 24cm$, $EC = 3cm$, and $BE$ and $AC$ are perpendicular. What is the length of the chord $BD$?
Here's what I've tried:
$ABE = 30$ which implies $AE = 12cm$ and therefore $B... | You are nearly there!
Note that angles CBD and CAD are equal. Angles BAD and EBC are then easily proved to be equal.
The right-angled triangle ABD is now similar to the triangle BEC of which you know all dimensions. Over to you?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3385957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $PSL_2(\mathbb Z)$ a Fuchsian group of the first kind? We know that as a discrete subgroup of $PSL_2(\mathbb R)$, $PSL_2(\mathbb Z)$ is a Fuchsian group. But how to prove/disprove that it is of the first kind. i.e. if every point on the extended real line is its limit point of some orbit?
| Hint: Prove that every rational point on the real line is fixed by a parabolic element of $PSL(2,{\mathbb Z})$. A sub-hint: Think first about stabilizers of nonzero elements of ${\mathbb Z}^2$ in $SL(2,{\mathbb Z})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3386065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding all integers $k \geq 2$ such that $k^2 \equiv 5k \pmod{15}$. What is going on here? The question is as follows:
Find all integers $k \geq 2$ such that $k^2 \equiv 5k \pmod{15}$.
I have an issue related to this question (its not about the solution to the question):
I know that $\overline{k} \in \mathbb{Z}_{15... | If $k^2\equiv5k\mod 15,$ then $3 $ and $ 5 $ divide $ k^2-5k=k(k-5)$,
so $3$ divides $k$ or $k-5$ and $5$ divides $k$ or $k-5$.
$5$ | $k$ iff $5$ | $k-5$, so we have $3$ divides $k$ or $k-5$ and $5$ divides $k$ and $k-5$.
That means $15$ divides $k$ or $k-5$; i.e., $k\equiv 0$ or $5 \mod 15$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3386183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Sum of squared eigenvalues is at most trace of adjoint product Specifically, I'm trying to solve the following:
Let $T$ be a complex $n\times n$ matrix. Let $\lambda_1,\cdots,\lambda_n$ be the eigenvalues of $T$, where each eigenvalue is repeated a number of times equal to its algebraic multiplicity. Prove that
$$\s... | Alternatively, we can use the Frobenius inner product: $\langle A,B\rangle=\operatorname{tr}(A^*B)$.
Replace $T$ with $T=U(D+N)U^*$ for some unitary $U$, diagonal $D$, and strictly upper triangular nilpotent $N$ (Schur's theorem). Then
\begin{align*}
\operatorname{tr}(T^*T)&=\langle T,T\rangle\\
&=\langle U(D+N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3386295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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